#### Degree Name

MS (Master of Science)

#### Program

Mathematical Sciences

#### Date of Award

5-2017

#### Committee Chair or Co-Chairs

Robert A. Beeler, Teresa Haynes

#### Committee Members

Robert Gardner, Rodney Keaton

#### Abstract

For a graph G = (V;E), a pebble distribution is defined as a mapping of the vertex set in to the integers, where each vertex begins with f(v) pebbles. A pebbling move takes two pebbles from some vertex adjacent to v and places one pebble on v. A rubbling move takes one pebble from each of two vertices that are adjacent to v and places one pebble on v. A vertex x is reachable under a pebbling distribution f if there exists some sequence of rubbling and pebbling moves that places a pebble on x. A pebbling distribution where every vertex is reachable is called a rubbling configuration. The t-restricted optimal rubbling number of G is the minimum number of pebbles required for a rubbling configuration where no vertex is initially assigned more than t pebbles. Here we present results on the 1-restricted optimal rubbling number and the 2- restricted optimal rubbling number.

#### Document Type

Dissertation - Open Access

#### Recommended Citation

Murphy, Kyle, "On t-Restricted Optimal Rubbling of Graphs" (2017). *Electronic Theses and Dissertations.* Paper 3251. https://dc.etsu.edu/etd/3251

#### Copyright

Copyright by the authors.